Brown–Forsythe test

Brown–Forsythe test is a statistical procedure developed by Robert Brown and Alan Forsythe. It is used to assess the equality of variances for a variable calculated for two or more groups. Unlike the traditional Levene's test, which relies on the mean to assess variance homogeneity, the Brown–Forsythe test uses the median, making it more robust to departures from normality. This characteristic makes the Brown–Forsythe test particularly useful in situations where the data may not follow a normal distribution.

Overview

The Brown–Forsythe test modifies the Levene's test by focusing on the median rather than the mean to measure the spread or variability among groups. This approach enhances its robustness against non-normal data distributions. The test is applicable in various fields, including medicine, psychology, and agriculture, where it's crucial to verify the assumption of equal variances across groups before performing certain types of statistical analyses, such as an Analysis of Variance (ANOVA).

Procedure

The Brown–Forsythe test involves several steps: 1. For each group, calculate the median of the data. 2. Compute the absolute deviations of each observation from the group's median. 3. Perform a one-way ANOVA on the absolute deviations. 4. The F-statistic from the ANOVA is used to determine if there is a significant difference in variances across groups.

Assumptions

The test assumes: - Independence of observations. - The groups are sampled from populations with identical shapes but possibly different variances.

Application

To apply the Brown–Forsythe test, researchers typically use statistical software packages that include this test as a function. The decision to reject or not reject the null hypothesis of equal variances is based on the p-value obtained from the test statistic. A small p-value (typically < 0.05) indicates that the variances are significantly different.

Advantages and Limitations

Advantages: - Robust against non-normal distributions. - Suitable for skewed data or data with outliers.

Limitations: - Like other variance tests, it can be sensitive to deviations from the assumption of identical distribution shapes. - May have less power than some alternatives when the data is normally distributed.

Example

Consider a study comparing the effect of three different diets on weight loss. The researcher could use the Brown–Forsythe test to check if the variance in weight loss across the three groups is equal before conducting an ANOVA to compare the mean weight loss.

See Also

- Levene's test - Analysis of Variance (ANOVA) - F-test - Statistical hypothesis testing